Design of an Image
Cryptosystem using Singular
Value Decomposition
Aseem Kumar Patel and Arup Kumar Pal
Department of Computer Science and Engineering
Bhubaneswar Institute of Technology
Bhubaneswar-752054, Orissa, India
Coverage of the Presentation
Introduction
Fundamentals of Image Compression,SVD and Secure Communication.
The proposed image cryptosystem (basedon SVD)
Experimental Results
Security Analysis
Conclusions
Introduction
Image Cryptosystem is useful to transmit importantimage through public channel.
It consists of mainly two parts Compression – Reducing network traffic
Encryption – Making data secret or confidential.
SVD is an efficient tool for image compression andsame has been used in our work.
Secure transmission of SVD compressed image usingcryptography techniques.
Image Compression Image Compression exploits various types of redundancies
present in the real images.
Two key factors are: Compression ratio
Tolerance in the quality degradation
Lossless compression Perfect reconstruction from compressed data (text documents,
medical images etc.)
Lossy compression Perfect reconstruction is not possible (is not essential) and we can
afford the partial loss in the image data as long as it is withintolerance.
4
Singular Value Decomposition (SVD)
Given any mn matrix M, algorithm to findmatrices U, D and V such that
Where U and V are column-orthogonal matrix and D is a diagonalmatrix.
Image Compression: Discard the trailinginsignificant singular values and keep the firstr (where r < k) singular values
5
Tm n m k k k n kM U D V
Tm n m r r r n rM U D V
Secure Communication
6
Encryption Key(K) Decryption Key(K)
Plain textCipher text
(Open Channel)
Enemy or
Adversary
Alice Encryption Decryption Bob
Plain text
Eve
7
The Proposed
Technique(Encoding)
Encoding
8
SVD
XOR
Key Formation
Proposed Algorithmic Steps for EncodingBeginStep 1: Decompose input secret image into three matrices U, D and
V respectively.Step 2: Select first r number of column vectors from each
decomposed matrix and discard rest of the column vectors.Step 3: Use truncated matrix, Vr obtained from Step 2 to generate a
key matrix.Step 4: Perform XOR operation between the produced key matrix
and the truncated matrix, Ur obtained from step 2.End
Decoding
10
XOR
Key Formation
T
r r rU D V
Algorithmic Steps (Decoding)
BeginStep 1: Generate a key matrix from the matrix Vr
using a secret PNS.Step 2: Perform XOR operation between the
produced key matrix and the matrix, EUr.Step 3: Reconstruct the image using the following
equationEnd
T
r r rU D V
12
Experimental Results
(b) Pepper Image(a) Lena Image
Original Test Images
Contd..(Experimental Results on Test Image Lena)
Decrypted ImageEncrypted image
Contd..(Experimental Results on Test Image Pepper)
Decrypted ImageEncrypted image
Security Analysis
The security of any encryption algorithm is measured bythe size of the key space.
In the proposed cryptosystem the secrecy depends on theselection of secret PNS.
In our experiment we have taken a PNS of length 65536.
The PNS guessing in our scheme is practically infeasiblebecause there will be one optimal PNS out of totalpossible 65536! PNSs.
Conclusions
In the proposed scheme, SVD is applied onthe secret image for compression andsubsequently the compressed files are partiallyencrypted for ensuring confidentiality.The proposed scheme reduces overallcomputation cost for encryption of the secretimage.The proposed scheme has been simulatedusing MATLAB and the satisfactory results havebeen found.
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Contd..
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